Duality Principle (Boolean Algebras)

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This proof is about the Duality Principle as applied to Boolean Algebras. For other uses, see Duality Principle.

Theorem

Let $\struct {S, \vee, \wedge}$ be a Boolean algebra.


Then any theorem in $\struct {S, \vee, \wedge}$ remains valid if both $\vee$ and $\wedge$ are interchanged, and also $\bot$ and $\top$ are interchanged throughout the whole theorem.


Proof

Let us take the axioms of a Boolean algebra $\struct {S, \wedge, \vee}$:

\((BA \ 0)\)   $:$   $S$ is closed under both $\vee$ and $\wedge$             
\((BA \ 1)\)   $:$   Both $\vee$ and $\wedge$ are commutative             
\((BA \ 2)\)   $:$   Both $\vee$ and $\wedge$ distribute over the other             
\((BA \ 3)\)   $:$   Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively             
\((BA \ 4)\)   $:$   $\forall a \in S: \exists \neg a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$             

It can be seen by inspection, that exchanging $\wedge$ and $\vee$, and $\bot$ and $\top$ throughout does not change the axioms.

Thus, what you get is a Boolean algebra again.

Hence the result.

$\blacksquare$


Sources